Hydrodynamic of reefs

7/25/2019 Hydrodynamic of reefs

1/21

Hydrodynamics of CoralReefs

Stephen G. Monismith

Environmental Fluid Mechanics Laboratory, Stanford University, Stanford,California 94305-4020; email: [email protected]

Annu. Rev. Fluid Mech. 2007. 39:3755

TheAnnual Review of Fluid Mechanicsis onlineat fluid.annualreviews.org

This articles doi:10.1146/annurev.fluid.38.050304.092125

Copyright c2007 by Annual Reviews.All rights reserved

0066-4189/07/0115-0037$20.00

Key Words

coastal engineering, biological fluid mechanics, water waves,geophysical fluid dynamics, environmental fluid mechanics,

turbulent flows, boundary layers

Abstract

The geometric complexity of coral reefs leads to interesting flui

mechanics problems at scales ranging from those of coral colonie

or even branches a few millimeters in diameter up to whole reef

that can be kilometers in horizontal extent. In many cases, both

at the colony and reef scale, unsteady flows, usually due to surfac

waves, behave very differently than do steady flows for which thcoral structures may appear to have quite high resistance to any

flow through their interior. Allowing for this difference, engineerin

formulae for mass transfer describe well the uptake of nutrients by

corals, although a priori determination of hydrodynamic roughnes

of corals and coral reefs is not yet possible. Surface wave-driven

flows are a common feature of many coral reefs and appear to follow

predictions of theories based on radiation stress gradients. Howeve

comparisons to observations have been relatively limited, and theris some question as to the role played by Stokes drift in these flows

Likeother near-shore environments, internal waves and flows drive

by horizontal buoyancy gradients can also be important.

37


2/21

INTRODUCTION

Coral reefs provide a wide and varied habitat that supports some of the most diverse

assemblages of living organisms found anywhere on Earth (Darwin 1842). Since

Odum & Odum (1955), it has been known that reefs represent islands of enormous

productivity becausethey areefficient at trapping nutrients, zooplankton,and possibly

phytoplankton from the surrounding waters (Yahel et al. 1998). This highlights the

primary connection between reef function and health and hydrodynamic processes.Recent work has highlighted how flow-mediated mass transfer may also prevent

coral bleaching (Nakamura & van Woesik 2001), the loss of symbiotic algae (thezooxanthallae) from the tissue of the coral polyps that can lead to death of the corals

(Hoegh-Guldberg 1999).

The hydrodynamics of coral reefs and coral colonies entails a wide range of scales

of fluid motions, starting with the largest scales characterizing flow such as eddies

produced by island wakes, smaller depth-scale turbulent features determined by reef

topography, and finally the smallest scales of flow, those comparable to single coral

colonies. I use this hierarchy of scales as a basis for organizing this review, starting

with the smallest scales and finishing at the largest scales. Due to space constraints,

important topics like the role of wave-induced coral breakage in shaping reefs, or thevalue of reefs as natural breakwaters, are left for others to review. In a like fashion, an

excellent review of work on the physical oceanography of reefs prior to 1990 can be

found in Andrews & Pickard (1990).

FLOW AT THE SCALE OF A CORAL COLONY

Most of the reef-building corals are modular organisms, i.e., each coral is a colony

consisting of numerous, interconnected, identical modules, called polyps. The coral

grows by extending its carbonate skeleton and adding more polyps. As Figure 1shows, this design concept is highly diverse, with some corals taking branching

forms, some foliated and some massive.

Given the geometric complexity of corals, it is somewhat more difficult to define

their size except perhaps in terms of the size of the box required to contain a given

colony. With this definition, the coral colony size can vary from 103 to 30 m3. Other

measures can also be defined, e.g., the ratio of the total surface area of the coral

to the plan-form area of the box that contains it. This ratio is important in that it

defines the amount of surface area of coral per unit area of reef across which mass

transfer can occur (Falter et al. 2005). One means to obtain this information, at leastfor smaller colonies, is to obtain a complete three-dimensional map of the colony by

X-ray tomography (Kaandorp et al. 2003). Using this approach, the area ratio is five for

the branching coral shown inFigure 2a(S. Chang, personal communication). This

colony structure allows for enhanced mass flux to the colony within limits discussed

below. In comparison, this ratio would be close to one for most engineered surfaces,

although the complex structures used to cool microprocessors must achieve a similar

enhancement to keep laptops like the one used to write this article from catching fire!

Most, if not all, of the work on colony-scale hydrodynamics has focused on

branching corals. Chamberlain & Graus (1975) carried out flow visualization studies

38 Monismith


3/21

Figure 1

Coral reef on the north shore of Moorea, F.P. (Photo by A. Santoro.)

Figure 2

(a) As seen by X-ray tomography:Stylophora pistillata, low velocity morphology (Chang et al.2004, Reidenbach et al. 2006a). (b) Flow inside the coral shown in (a) as measured by MagneticResonance Velocimetry (MRV). This is a horizontal plane taken near mid-height using theMRV technique of Elkins et al. (2003). Image courtesy of S. Chang and C. Elkins.

www.annualreviews.org Hydrodynamics of Coral Reefs 39


4/21

using both real coral specimens and arrays of cylindrical rods, finding that much ofthe flow that approaches branching corals is diverted around and over the coral. As

expected, coral geometries with denser branching tend to divert more flow to the

exterior, whereas coral geometries with sparser branching allow more flow through

the interior (Reidenbach et al. 2006a). Examining flows inside the coral colony has

proved difficult since the outer branches block both optical and acoustic access to

the interior. However, recent work (Elkins et al. 2003) (Figure 2b) using magnetic

resonance velocimetry (MRV), which is not obstructed by the solid matrix, reveals

the complex details of the interaction of wakes from upstream branches encountering

downstream branches that leads to blocking of interior flows (S. Chang & C. Elkins,unpublished data).

The flow through the coral colony is different in the presence of waves. In exper-

iments with several species and morphologies of branching coral, Reidenbach et al.

(2006a) confirmed the steady flow behavior discussed above by looking at the dissolu-

tion of small gypsum plugs (a proxy for velocity) placed throughout the coral colony

interiors. When waves were added, the velocities inside the colony were inferred to

be similar to those outside the colony, leading to a wave enhancement of interior

mass transfer. Lowe et al. (2005b,c) confirmed this behavior for arrays of cylindersserving as a simple model of a coral, showing that the enhancement depends on the

Keulegan-Carpenter number (Dean & Dalrymple 1991),

Kc =UwS

, (1)

where Uwis the waveorbitalvelocity, is the frequency, and S is the cylinderspacing.

Kc represents the ratio of the orbital excursion to object geometry, in this case the

cylinder spacing. When Kc is large, the flow is drag dominated and velocities inside

the colony are much lower than in the free stream. In contrast, when Kc is small,the interstitial flow is inertia dominated and interior velocities more nearly match

exterior ones, and total mass transfer is enhanced over that of steady flows.

Unlike engineered structures, corals are living structures that respond to flow in

complex ways that can alter their physical structure. For example, flow variations

inside the colony may induce localized calcification and a specific preferential growth

form (Kaandorp et al. 2003, 2005; Kaandorp & Kubler 2001; Lesser et al. 1994),

or may lead to branch orientation that optimizes nutrient uptake or prey capture

(Helmuth & Sebens 1993). Veron & Pichon (1976) show examples of the resulting

flow-related intraspecific plasticity variability. Amatzia Genin of the Steinitz Labin Israel has carried out simple experiments demonstrating the effect of flow on

morphology in which coral of a given species and flow environment is exposed in situ

to higher flow velocities using underwater pumps (see Reidenbach et al. 2006a). The

effect of this flow change is to make the coral skeleton almost uniformly thicker, thus

reducing the size of the spaces between the branches.

For branching corals this connection should be complicated by the fact that the

distribution and rates of mass transfer for flow through and over corals depends on the

details of separation from the branches and subsequent reattachment on downstream

branches. Separated and reattached flows cause large variations in local heat (and

40 Monismith


5/21

mass) transfer such that heat transfer and momentum can be decoupled (Vogel &Eaton 1985). Using lattice Boltzmann methods to solve the complicated interior

flow problem, and assuming that local growth is limited by diffusion to the structure

(diffusion limited accretion), Kaandorp and colleagues (Kaandorp et al. 2003, 2005;

Kaandorp & Kubler 2001) predict structures that are quite similar in appearance to

branching corals.

However, although the qualitative appearance is correct, there is no doubt that the

devilis in the details.For example,to correctlypredict mass transfer of nutrients to the

corals, the thin diffusive boundary layers on the coral branches must be resolved. For

most nutrients of interest, which have Schmidt numbers Sc = /D that are O(1000),scalar boundary layers can be expected to be a factor of Sc 1/3 10 times smaller than

any local viscous boundary layers (see e.g., Kays & Crawford 1993). For example,

for a 1 cm diameter (d) branch with a flow (U) of 10 cm/s, the viscous boundary

layer thickness will bev (d/U)1/2 0.03 cm, and so a numerical model of mass

transfer to this branch would need to resolve layers that are 3 103 cm thick. Thus,

a grid for a typical coral colony with a linear dimension of 10 cm would have of the

order of 30003 1010 grid points. Even with immersed boundary methods (Chang

et al. 2004, Iaccarino & Verzicco 2003) this is still out of reach, although it does seem

that computing mass transfer for Sc 1, which would only require 107 grid points,may be attainable and interesting.

However, there is also biology to consider! The fact that the observed change in

branch thickness with increased flow is nearly uniform suggests the possible impor-

tance of internal translocation of nutrients, photosynthates, etc. in the tissue layer

that connects the polyps of a given colony (Rinkevich & Loya 1983). Were there no

translocation, presumably the geometric modifications would be entirely localized,

as modeled by Kaandorp and colleagues. Thus, it seems that to produce a virtual

coral would require coupling the challenging flow model with appropriate spatiallyexplicit models of the biology of the polyps. In any case, although a convincing de-

termination of the mechanism(s) involved remains to be produced, the fact that flow

velocity affects morphology seems incontrovertible.

BOUNDARY LAYER FLOW OVER REEFS:FLOWS AT THE 1- TO 10-M SCALE

At scales of 1 to 10 m, the most obvious physical feature of coral reefs is that they are

remarkably rough, having bottom drag coefficients, CD, that are typically ten timeslarger (or more) than the canonical value of 0.0025 found for muddy or sandy sea

beds (Lugo-Fernandez et al. 1998b, Roberts et al. 1975). Baird & Atkinson (1997)

report values of equivalent sand grain roughness, ks = 0.28 m, for flows in a flume

filled with coral skeletons, although McDonald et al. (2006) found that, in part, this

large apparent roughness was due to the fact that laboratory experiments in flows of

limited depth force significant amounts of flow through the corals rather than over it

as with deeper flows. Measurements of wave damping over the reef flat at Kaneohe

Bay reported in Falter et al. (2004) and Lowe et al. (2005a), as well as measure-

ments of wave setup reported in Tait (1972) and Gerritsen (1980), gave similar values.



6/21

Reidenbach et al. (2006) found that ks = 1m for nonwavy flow over the fringing reef inEilat.

As is common in the study of geophysical boundary layers (see e.g., Lueck &Lu

1998), these values of ks are essentially determined by choosing ks, so that observed

velocity profile u(z) best fits the law of the wall

u(z) =u

ln30z

ks . (2)Here= 0.41 is the von Karman constant and z is the height above the bed. At least

for steady flows, this appears to be justified: Reidenbach et al.s (2006b) simultaneousmeasurements of velocity profiles and turbulent stresses at several heights above the

bedmatchthelawofthewall,aswellasindicatethattherateofdissipationofturbulent

kinetic energy is nearly in balance with its rate of production, as would be expected

in a canonical rough-wall turbulent boundary layer. However, although there have

been various approaches to directly measuring the real roughness of reefs, e.g., by

draping chains or measuring a dense array of heights, there is currently no accepted

way of measuring reef roughness that can be translated into values of CDor ks.

The challenge of predicting drag on coral reefs derives from their geometric

complexity and how the colony-scale flows combine with the effects of other nearbyroughness (rocks, sponges, etc.). Drawing the analogy to canopy flows (Ghisalberti

& Nepf 2004, Raupach & Thom 1981), particularly urban canopy flows (see Britter

& Hanna 2003, Coceal & Belcher 2004), Lowe et al. (2005b) show how the drag

on an array of cylinders, a crude model of a coral reef, depends on the ratio of the

spacing, S, of the cylinders to their diameter, D. For steady flows, this leads to good

predictions of the overall drag on the array, at least for the range of cylinder spacing

(1


7/21

k+=uks/,and(b) nutrients of interest have large values of Sc, i.e., O(1000). Lackingappropriate data, Bilger & Atkinson (1992) combined the high k+ but low Sc (Sc= 7)

results of Dipprey & Sabersky (1963) with the low k+ but high Sc results of Dawson

& Trass (1972). Initially, it appeared that mass transfer was much greater than would

be expected based on flow over rough walls (Atkinson & Bilger 1992). However, later

work (Falter et al. 2004, 2005) that accounts for the fact that waves are the dominant

motion on the reefs and in the tanks that Atkinson and colleagues studied shows that

the mass transfer relation (shown in Figure 3)

St=m

U0C 0=

C0.5D(k+)0.2 Sc0.6

, (3)

where m is the mass flux per unit area, St is the Stanton number, C0is the free streamnutrient concentration, U0 is the free stream velocity = the wave orbital velocity

when waves are dominant, andis an O(1) constant, fit all of the mass transfer data

both from real reefs and from coral assemblages located in flumes (Baird & Atkinson

1997, Thomas & Atkinson 1997).

Gypsumcoated skeletonsOvergrown highrelief rubble

Overgrown lowrelief rubble

Porites compressa

Pocillopora damicornis

0 1.0 2.0 3.00

0.4

0.8

1.2

1.6

(x10-4m s-1)

Rek

0.2Sc

i

0.58

U cf/2

Smeas

(x10-4ms

-1)

Figure 3

Mass transfer to coral colonies and reefs measured by Falter et al. (2004) and shown in theirfigure 1. Note that in this plot Rek= k

+ and that the plot is given in dimensional terms. The

filled symbols represent experiments with gypsum-coated skeletons whereas the open symbolsrefer to ammonium uptake by experimental reef communities consisting of high-relief coralrubble overgrown with turf and macroalgae (squares), low-relief coral rubble overgrown withturf and macroalgae (circles),Porites compressa(triangles), andPocillopora damicornis(diamonds).Please see Falter et al. (2004) for an explanation of the symbols. Copyright 2004 by theAmerican Society of Limnology and Oceanography.



8/21

There are three caveats to this fit: (a) The surface area available for mass transferper unit area of reef must be accounted for. As discussed above, this ratio may be five

or more for dense coral canopies (Falter et al. 2005). (b) The value of CDin the wavy

case is the one required to match the observed damping of waves (Falter et al. 2004,

Hearn et al. 2001). (c) This implies that nutrient uptake is linked directly to the wave

climate of the reef; there is still a factor 1.3.5 enhancement due to oscillatory motion

that has been included (Falter et al. 2005, Lowe et al. 2005c, Reidenbach et al. 2006a).

The factor of (k+)0.2 appearing in Equation 3 implies that as roughness increases,

mass transfer is reduced. This factor represents the sheltering of lower parts of the

surface by the higher parts of the roughness elements, i.e., that velocities and hencemass transferare higher nearthe tops of the roughnessthanin the troughs between the

roughness elements (Dipprey & Sabersky 1963). Given the high interior resistance of

branching corals to flow, this model seems applicable to both isolated coral colonies

as well as continuous canopies of coral, although for waves, the velocity enhancement

seen in laboratory experiments may mean that the constant of proportionality in

Equation 3 may also depend on an appropriately defined value of Kc, as does the drag

(Lowe et al. 2005c).

Turbulence produced by the coral reef also affects the rate particulates suspended

in the water column that can be grazed by the overall reef community (Frechetteet al. 1988, Genin et al. 2002). Because rougher boundaries lead to higher rates

of mass transfer, they also make grazing more efficient. It seems possible that as a

reef grows and becomes rougher, it can support denser assemblages of other benthic

organisms, e.g., sponges or boring bivalves, i.e., richer benthic communities that add

to the diversity of life on the reef. On the other hand, in nonwavy flows, individual

colonies may see diminishing returns as they become progressively more shaded

by their neighbors. The hypothesis that these two effects can come into balance on

some reefs is an interesting one that deserves an experimental test (A. Genin, personalcommunication).

REEF-SCALE FLOWS: FLOWS AT SCALEOF 100 M TO 1000 M

Wave-Driven Flows over Coral Reefs

In many coral reef lagoons, surface wavedrivenflows often (but notalways) dominate.

Of allareas of research connectingcorals andhydrodynamics, thetopic of wave-drivenflows has probably received the most attention, beginning with pioneering studies

by Von Arx (1954) and Munk et al. (1949), which consider flushing of reef lagoons

by wave-driven flows over the fringing reef to more recent work examining wave

transformation and wave forcing of flows over and along fringing reefs (Andr efouet

et al. 2001; Angwenyi & Rydberg 2005; Gourlay & Colleter 2005; Hardy & Young

1996; Hearn 1999; Kraines et al. 1998, 1999; Lowe et al. 2005a; Lugo-Fernandez

et al. 1998a,b; Prager 1991; Roberts et al. 1975; Roberts & Suhayda 1983; Symonds

& Black 2001; Symonds et al. 1995; Tartinville & Rancher 2000; Young 1989). The

basic idea (Munk & Sargent 1948, Young 1989) is that incident waves break on the

44 Monismith


9/21

Lagoon

FlowSurfzone

Br

eakpoint

Waves

Wave setup Shore

Reef flat

Forereef

(x)

h(x)

Figure 4

Definition sketch for wave-driven flow over a reef and into a lagoon. (Redrawn from Hearn1999; courtesy of R. Lowe.)

offshore face of the reef and push water onto the reef flat and then into the lagoon (if

there is one; seeFigure 4).Symonds et al. (1995)and Hearn (1999)present observations showing this connec-

tion. For example, Hearn (1999) found (with significant scatter) that induced cross-

reef currents were linearly proportional to offshore wave height. In a remarkable

series of lab experiments studying wave-driven flows over reefs, Gourlay (1996a,b)

varied both mean water level relative to the reef flat level incident wave height, find-

ing two regimes of flow behavior: (a) reef-top control: When the mean water level

is sufficiently far above the reef flat, the reef flat flow is in a friction-pressure gra-

dient balance; (b) reef-rim control: When the mean water level is at or below the

(raised) reef flat, a hydraulic control at the lagoon end of the flat determines thedepth of the overall flow. In the latter case, wave breaking occured on the fore-reef

and the wave-driven flow was due to swash running up and onto the reef flat. Note

that in Gourlays experiments the reef flat was relatively smooth (no roughness ele-

ments) in these experiments, and so the rim-control flows may be less relevant to real

reefs.

All models of wave-driven flows over reefs (e.g., Gourlay & Colleter 2005, Hearn

1999, Symonds et al. 1995) are basedon the idea first advanced by Longuet-Higgins &

Stewart (1962)that spatialgradients in wave radiation stress, i.e., the depth-integratedmomentum flux (including pressure forces) associated with propagating waves, ap-

pears as a body force acting on the mean (wave averaged marked with an overbar)

flow. Thus, they should be quite similar in dynamics to longshore flows on beaches

forced by incident waves (Lentz et al. 1999). As presented in Mei (1989), including

radiation stresses, the momentum equations governing the horizontal flow can be

written as

Ujt + Ui

Ujxi

= g

xj

1

(+ h)

Sijxi

Bj

(+ h), (4)



10/21

where Ui = UEi +USi is the ith component of the Lagrangian mean velocity, written

here as the sum of the depth-averaged Eulerian mean velocity and depth-averaged

Stokes drift velocity (Longuet-Higgins 1969), is the deviation of the free surface

from mean sea level (MSL), h is the depth of the bottom below MSL, Sij is theradiation stress tensor (given below), and Bi is the mean (wave-averaged) bottom

stress. For waves propagating in the x1direction (normal to shore) with amplitude A,

wave number k, Longuet-Higgins & Stewart (1962) show that

S11 =gA2

4

1 +

4kh

sinh(2kh)

. (5)

Thus, as waves shoal (changing kh), there will be variations in S11that may drive

currents (Longuet Higgins 1970) or result in the setupof the free surface (Bowen et al.

1968). Note that Equation 4 applies to Lagrangian, not Eulerian, mean velocities.Although the contribution of the Stokes drift to the overall flow may be negligible

for longshore flows on a beach, this may not be true for flows over reefs in which the

mean Eulerian flow and Stokes drift are coaligned.

Symonds et al. (1995) were the first to apply these ideas to computing wave-forced

flow on reefs, analyzing the case of a simple geometry in which the offshore sectionwas represented as a plane beach while onshore of the break was a constant depth reef

flat. Waves were assumed to shoal and then break when the critical breaking criterion

A=h/2, where the constant 0.8 was derived from experiments examining

waves on beaches. In this model, offshore of the break, there is a set down of thewater surface, because the approximate momentum balance reads (x1directed positive

offshore)

g

x1

1

(+ h)

S11x1

, (6)

withS11/x1 < 0. Within the breaker zone itself, there is a setup of the free surface

since Equation 6 still holds (c.f. Hearn 1999), but with S11/x1> 0. If there were

no reef flat, then there would be a setup of the water surface just like on beaches(Gerritsen 1980, Tait 1972).

Inshore of the break the water surface slopes down from its high point at the endof the break to the water level in the lagoon, which is assumed to be the same as

offshore, driving a flow with

g

x1

Bj

(+ h)=

CDu |u|

(+ h) r

U

h. (7)

Here the drag is expressed as the average of the instantaneous bottom friction, in-

cluding both mean flow and waves or, for analytical convenience, using a linearized

friction model (Hearn 1999, Symonds et al. 1995). Because the waves are assumed not

to change in this region (not always a good assumption; see Lowe et al. 2005a), there

is no radiation stress gradient on the reef flat. Like most circulation models, Equa-

tion 7 is closed by choosing bottom friction coefficients to best match observations.

These will be relatively large owing to the roughness of the reef (Lugo-Fernandez

et al. 1998b) and will be a function of the wave orbital velocity (Hearn 1999). Again,

46 Monismith


11/21

whatever the roughness, the subtle point that the model predicts Lagrangian meanvelocities complicates things because the friction formulations are expressed in terms

of the Eulerian means.

Model results from Symonds et al. (1995) compare well with currents measured on

John Brewer reef, a small reef located in the Great Barrier Reef complex, although

significant differences, possibly attributable to other forcing, remain (Symonds &

Black 2001). Lugo-Fernandez et al. (1998b) use a model similar to that of Symonds

et al. to successfully hindcast sea level changes on Tague Reef in the Caribbean.

Hearns (1999) comparisons are more limited, for example citing one observation

from Kaneohe Bay, Hawaii. In contrast, Gourlay & Colleters (2005) modeling doesan excellent job of matching Gourlays earlier observations. In this case, it is evident

that the control offered by laboratory experiments usefully isolates the effects of

geometry on wave-driven flows.

Hearns analysis points out an important aspect of wave-driven flows over reefs:

When the depth over the reef is shallow, flows will be weak because friction is strong,

whereas when the depth over the reef is large, flows will be weak because of limited

breaking and hence small radiation stress gradients (see also Gourlay & Colleter

2005). Thus, there will be a depth at which the wave-driven flow is a maximum.Moreover, tidal variations in depth will produce tidal variations in mean flows over

the reef.

Although useful in their simplicity, the assumptions made by Symonds et al. and

Hearn. may not always be valid. For example, both models assume that is zero at

the end of the reef flat, whereas, in reality, the water level must rise in the lagoon to

drive flow out through passes in the reef back to the ocean. These additional features,

e.g., tides, setup in the lagoon, radiation stresses on the reef flat, are easily dealt with

in two- and three-dimensional circulation models. To date, these models have gen-

erally used relatively simple wave models, i.e., ones that include refraction and wavebreaking only, and used the resulting wave field to calculate radiation stress gradi-

ents to force mean flows (Kraines et al. 1998, 1999; Prager 1991; Symonds & Black

2001; Tartinville & Rancher 2000). In both two-dimensional and three-dimensional

models (Tartinville et al. 1997), the radiation stress gradient is applied as a body

force. The comparison between observations (which have generally been limited)

and modeling looks promising, and has permitted estimation of important biogeo-

chemical hydrodynamic measures such as Lagoon residence time (Tartinville et al.

1997). More sophisticated approaches to predicting wave dynamics, e.g., Boussinesq

wave models (Skotner & Apelt 1999) or models accounting explicitly for effects oflarge bottom slope on wave propagation (Massel & Gourlay 2000), have also been

used, albeit only to look at the waves alone.

Nonetheless, the bottom line is that, unlike the case for beaches (Lentz &

Raubenheimer 1999), measurements of mean free surface changes have not been

done in conjunction with current measurements as would be needed to definitively

test any of the models discussed above. Conversely, given the issue of Eulerian (what

is measured) and Lagrangian (what is predicted) means, it seems that observations

of waves and wave-driven flows on coral reefs might also lead to new insights intofundamental aspects of the way waves and mean flows interact.



12/21

Flows Driven by Tides and Buoyancy

Much of the work on large-scale reef hydrodynamics has focused on the role reef

geometry plays in determining patterns and rates of horizontal transport, particularly

that of the larvae of coral reef fish and other organisms (Black 1993, Cowen et al.

2000). These larger-scale processes are only slightly, if at all, influenced by waves.

Wolanski and colleagues at the Australian Institute of Marine Sciences have focused

on inter-reef transport in the Great Barrier Reef complex, showing how flows in thisregion reflect the separate influences of shelf waves (Wolanski & Bennett 1983) and

tides (Wolanski 1983). One of their more general results is the importance of reefroughness in generating large-scale vorticity. Field observation, laboratory work, and

computation(see Wolanski et al. 1994) clarify that flowaroundisolated reefs generates

substantial vertical vorticity organized as large, reef-scale eddies, more or less like

those one sees shed by any bluff body. However, several twists emerge in shallow

water: Most notably, over the bottom of the surrounding ocean these eddies develop

secondary flows formally analogous to those that arise through Ekman layer dynamics

(Wolanski et al. 1984).

On the basis of Wolanskis work, we might expect that less dramatic variations

in topography associated with larger-scale patchiness of the reefs will likewise leadto the generation of vertical vorticity and hence of more complicated secondary

flows, particularly those involving onshore-offshore exchanges. For example, if the

longshore flow encounters a region of high roughness, it should diverge offshore;

if it encounters a region of low roughness it should accelerate and entrain offshore

fluid. As an extreme example of this effect, spectacular residual flows have been seen

in lagoon-atoll reef systems like that at Enewetak (Atkinson et al. 1981), where the

channels through the fringing reef can be the site of remarkably strong tidal currents.

Pointing to the strong coupling of hydrodynamic and biochemical processes, aswell as to the importance of oceanic stratification, Leichter et al. (1996, 2003) observed

significant transport of nutrients by internal waves from deep water onto Conch

Reef on the Southeast Florida Shelf. These observations showed large-amplitudeinternal waves generated tidally that propagated onshore, shoaled, and broke. The

details of the generation mechanism are currently unclear, but there is evidence that

Leichter et al. may be observing the nonlinear near-shore behavior of the cross-

Florida strait internal seiche (Niiler1968). Besides transporting nutrients,these waves

also move cold water up the shelf; the resulting periodic cooling events may help to

forestall bleaching during times when water temperatures might otherwise exceedthe bleaching threshold (Riegl & Piller 2003).

The steep topography of reefs can also lead to baroclinic exchange flows through

the mechanisms known as differential heating or cooling (Monismith et al. 2006,

Symonds & Gardiner-Garden 1994). When heated (by the sun) or cooled (mostly by

evaporation), the shallow near-shore waters of the reef will heat or cool more rapidlythan deeper waters offshore, leading to onshore-offshore temperature gradients and,

hence, baroclinic pressure gradients. Neiman et al. (2004) observed gravity currents

produced bycooling ofthe reef off Aqaba onthe easternshore of the northernRed Sea,findingthat these could result in thesignificantexport into the phytoplankton-derived

48 Monismith


13/21

carbon in the deeper waters of the Gulf of Aqaba. Monismithet al. (2006) documentedboth heating and cooling flows on the Eilat reef, finding that the momentum balance

is mainly steady and inertial and advective heat transport are dominant (Sturman et al.

1999), although bottom friction and unsteadiness in the thermal forcing can also be

important. The inertial scaling gives a sheared cross-shore flow,V, that varies as

V 1/3

g

H h

cp 1/3

, (8)

where is the bottom slope, is the thermal expansivity,H is the surface heatflux, h is the local depth, is the fluid density, and cp is the heat capacity. The

effects of unsteadiness and bottom friction are secondary, generally acting to weaken

the exchange flow, especially when the longshore flow is strong. Like internal waveupwelling, one can speculate that these convective flows provide a mechanism for

locally reducing the thermal stress that canlead to bleaching (Hoegh-Guldberg 1999),

as well as enhancing physical connectivity between the reef and ocean.

Although it has received scant attention, the flow of stratified fluid over a reef

should be expected to produce relative strong vertical mixing for even relatively slow

mean flows, meaning that reefs may be good sites for studying bottom mixed-layerdynamics, including sloping boundary layer dynamics (Garrett et al. (1993), or turbu-

lence dynamics in stratified fluids. Oneof theclearest observations of boundary mixing

in the ocean is that reported in Wolanski (1987) for flow in the lee of a coral island.

SUMMARY

Corals and coral reefs exemplify organisms and ecosystems that are shaped by flow.

At the smallest scales, the hydrodynamics of flow through and over very complicated

bodies determines local mass transfer and, in ways yet to be determined, morphology

of the coral skeleton. Seen at larger scales, the carbonate framework that coral polyps

manufacture appears to be a very rough surface, one that promotes mass transferbetween the overlying water andthe organisms of the reef, significantly damps surface

waves, and, of course, provides a rich habitat for a wide variety of organisms. At

the scale of the reef as a whole, coral reefs are similar to other near-shore coastal

environments, e.g., sandy beaches or kelp forests on rocky shores, in that flows can

be driven by various combinations of tides, buoyancy, and internal and surface waves.

Remarkably, it appears that reefs can be excellent (and pleasant!) laboratories in which

to study these processes. Because the geometries are different from those of more

commonly studied systems, i.e., beaches, work on reefs should be expected to yieldnew insights into the dynamics of coastal flows more generally.

ACKNOWLEDGMENTS

ItisapleasuretoacknowledgethefriendsandcolleagueswithwhomIhavebeenlearn-

ing about corals and reefs these past few years. I am indebted to Marlin Atkinson and

Amaztia Genin, who have shared with me their remarkable interests in understanding



14/21

how reefs work, and to Jim Hench and Kristen Davis, who have encouraged me tosee for myself how reefs look underwater. Much of the material discussed represents

collaborations between our group at Stanford and colleagues at other institutions.

In addition to the people mentioned above, this group of coral reefers includes Jeff

Koseff, Matt Reidenbach, Ryan Lowe, Sandy Chang, Johanna Rosman, Jim Falter,

Geno Pawlik, Roi Holtzman, Gitai Yahel, Uri Shavit, Jim Leichter, Chris Martens,

Niels Lindquist, Gianlucca Iaccarino, John Eaton, Chris Elkins, and Adina Paytan. I

am grateful to NSF for their support through grants OCE-0117859, OCE-0452800,

and OCE-0425312. Additional funding was provided by the Israel-U.S. Binational

Science Foundation, the National Undersea Research Center at the University ofNorth Carolina Wilmington (project SEGM-2004-20A), the Stanford Bio-X pro-

gram, and the Singapore-Stanford Partnership.

LITERATURE CITED

Andrefouet S. Pages J, Tartinville B. 2001. Water renewal time for classification of

atoll lagoons in the TuamotuArchipelago(French Polynesia). Coral Reefs20:399

408

Andrews JC, Pickard GL. 1990. The physical oceanography of coral reef systems. In

Ecosystems of the World: Coral Reefs, ed. Z. Dubinsky, 1145. Amsterdam: Elsevier.

550 pp.

Angwenyi CM, Rydberg L. 2005. Wave-driven circulation across the coral reef at

Bamburi Lagoon, Kenya.Estuarine Coastal Shelf Sci. 63:44754

Atkinson M, Smith SV, Stroup ED. 1981. Circulation in Enewetak Atoll lagoon.Proc.

Fourth Intl. Coral Reef Symp. 1:33538

Atkinson MJ. 1987. Rates of phosphate uptake by coral reef flat communities.Limnol.

Oceanogr. 32:42635Atkinson MJ, Bilger RW. 1992. Effects of water velocity on phosphate uptake in coral

reef-flat communities.Limnol. Oceanogr. 37:27379

Atkinson MJ, Falter JL, Hearn CJ. 2001. Nutrient dynamics in the Biosphere 2 coral

reef mesocosm: water velocity controls of NH4 and PO4 uptake. Coral Reefs

20:34146

Baird ME, Atkinson MJ. 1997. Measurement and prediction of mass transfer to ex-

perimental coral reef communities.Limnol. Oceanogr. 42:168593

Bilger RW, Atkinson MJ. 1992. Anomalous mass-transfer of phosphate on coral reefs.

Limnol. Oceanogr. 37:26172Black KP. 1993. The relative importance of local retention and inter-reef dispersal

of neutrally buoyant material on coral reefs.Coral Reefs12:4453

Bowen AJ, Inman DL, Simmons VP. 1968. Wave setdown and setup.J. Geophys. Res..

73:256977

Britter RE, Hanna SR. 2003. Flow and dispersion in urban areas. Annu. Rev. Fluid

Mech. 35:46996

Chamberlain JA, Graus RR. 1975. Water flow and hydromechanical adaptations of

branched reef corals.Bull. Marine Sci. 25:112125

Chang S, Iaccarino G, Elkins C, Eaton J, Monismith S. 2004. A computational and

50 Monismith


15/21

experimental investigation of flow inside a branched coral colony. In Center for

Turbulence Research Annual Research Briefs 2004, ed. P. Moin 4554. Stanford:

Stanford Univ.

Coceal O, Belcher SE. 2004. A canopy model of mean winds through urban areas.

Q. J. Roy. Met. Soc. 130:134972

Cowen RK, Lwiza KMN, Sponaugle S, Paris CB, Olson DB. 2000. Connectivity of

marine populations: open or vlosed?Science287:85759

Darwin CH. 1842. The Structure and Distribution of Coral Reefs. London: Smith, Elder

and Co. 207 pp.

Dawson DA, Trass O. 1972. Mass transfer at rough surfaces. Int. J. Heat Mass Trans.15:131736

Dean RG, Dalrymple RA. 1991. Water Wave Mechanics for Engineers and Scientists

(2nd ed.). Singapore: World Sci. 353 pp.

Dipprey DF, Sabersky RH. 1963. Heat and momentum transfer in smooth and rough

tubes at various Prandtl numbers.Int. J. Heat Mass Trans. 6:32953

Elkins C, Eaton JK, Markl M, Pelc N. 2003. 4D magnetic resonance velocimetry for

mean velocity measurements in complex turbulent flows.Exp. Fluids34:494503

Falter JL, Atkinson MJ, Merrifield M. 2004. Mass-transfer limitation of nutrientuptake by a wave-dominated reef flat community.Limnol. Oceanog. 49:182031

Falter JL, Atkinson MJ, Coimbra CFM. 2005. Effects of surface roughness and os-

cillatory flow on dissolution of plaster forms: evidence for nutrient mass transfer

to coral reef communities.Limnol. Oceanogr. 50:24654

Frechette M, Butman CA, Geyer WA. 1988. The importance of boundary layer flows

in supplying phytoplankton to the benthic suspension feeder. Mytilus Edulis.

Limnol. Oceanogr. 34:1936

Garrett CJR, McCready P, Rhines P. 1993. Boundary mixing and arrested Ekman

layers: rotating stratified flow near a sloping boundary. Annu. Rev. Fluid Mech.25:291323

Genin A, Yahel G, Reidenbach MA, Monismith SG, Koseff JR. 2002. Intense ben-

thic grazing on phytoplankton in coral reefs revealed using the control volume

approach.Oceanography15:9097

Gerritsen F. 1980. Wave attenuation and wave set-up on a coastal reef.Proc. Int. Conf.

Coastal Eng., 17th, Sydney, Australia,1:44461

Ghisalberti M, Nepf HM. 2004. Limited growth of vegetated shear layers. Water

Resources Res. 40:W0750.2. doi:10.1029./2003.WR002776

Gourlay MR. 1996a. Wave set-up on coral reefs. 1. Set-up and wave-generated flowon an idealised two dimensional reef.J. Coastal Eng.27:16193

Gourlay MR. 1996b. Wave set-up on coral reefs. 2. Wave set-up on reefs with various

profiles.J. Coastal Eng.28:1755

Gourlay MR, Colleter G. 2005. Wave-generated flow on coral reefs: an analysis for

two-dimensional horizontal reef-tops with steep faces.J. Coastal Eng. 52:35387

Hardy TA, Young I. 1996. Field study of wave attenuation on an offshore coral reef.

J. Geophys. Res.101: 1431126

Hearn CJ. 1999. Wave-breaking hydrodynamics within coral reef systems and theeffect of changing sea level.J. Geophys. Res. 104:3000719



16/21

Hearn CJ, Atkinson MJ, Falter JL. 2001. A physical derivation of nutrient-uptakerates in coral reefs: effects of roughness and waves.Coral Reefs20:34756

HelmuthB, Sebens KP. 1993. Theinfluence of colony morphology andorientation to

flow on particle capture by the scleractinian coralAgaricia agaricites(Linnaeus).

J. Exp. Mar. Biol. Ecol. 165:25178

Hoegh-Gulberg O. 1999. Climate change, coral bleaching and the future of the

worlds coral reefs.Mar. Freshwater Res. 50:83966

Iaccarino G, Verzicco R. 2003. Immersed boundary technique for turbulent flow

simulations.Appl. Mech. Rev. 56:33147

Kaandorp JA, Kubler JE. 2001.The Algorithmic Beauty of Seaweeds, Sponges and Corals.Berlin: Springer Verlag. 193 pp.

Kaandorp JA, Koopman EA, Sloot PMA, Bak RPM, Vermeij MJA, Lampmann LEH.

2003. Simulation and analysis of flow patterns around the scleractinian coral

Madracis mirabilis (Duchassaing and Michelotti). Phil. Trans. R. Soc. London B.

358:155157

Kaandorp JA, Sloot PMA, Merks RMH, Bak RPM, Vermeij MJA, Maier C. 2005.

Morphogenesis of the branching reef coralMadracis mirabilis.Proc. R. Soc. London

B272:12733Kays WM, Crawford ME. 1993.Convective Heat and Mass Transfer,3rd ed. New York:

McGraw-Hill. 601 pp.

Kraines SB, Yanagi T, Isobe M, Komiyama H. 1998. Wind-wave driven circulation

on the coral reef at Bora Bay, Miyako Island. Coral Reefs17:13343

Kraines SB, Suzuki A, Yanagi T, Isobe M, Guo X, Komiyama H. 1999. Rapid water

exchange between the lagoon and the open ocean at Majuro Atoll due to wind,

waves, and tide.J. Geophys. Res. 104:156356653

Leichter JJ, Wing SR, Miller SL, Denny MW. 1996. Pulsed delivery of subther-

mocline water to Conch Reef (Florida Keys), by internal tide bores. Limnol.

Oceanogr. 41:14901501

Leichter JJ, Stewart HL, Miller SL. 2003. Episodic nutrient transport to Florida coral

reefs.Limnol. Oceanogr. 48:1394407

Lentz S, Raubenheimer B. 1999. Field observations of wave setup. J. Geophys. Res.

104: 2586776

Lentz S, Guza RT, Elgar S, Feddersen F, Herbers THC. 1999. Momentum balances

on the North Carolina inner shelf.J. Geophys. Res. 104:1820526

Lesser MP, Weis VM, Patterson MR, Jokiel PL. 1994. Effects of morphology and

water motion on carbon delivery and productivity in the reef coral,Pocillopora

damicornis (Linnaeus): diffusion barriers, inorganic carbon limitation, and bio-

chemical plasticity.J. Exp. Mar. Biol. Ecol. 178:15379

Levinton JS. 1995.Marine Biology. New York: Oxford Univ. Press. 420 pp.

Longuet-Higgins MS. 1969. On the transport of mass by timevarying ocean currents.

Deep-Sea Res. 16:43247

Longuet-Higgins MS. 1970. Longshore currents generated by obliquely incident sea

waves.J. Geophys. Res. 75:677889

Longuet-Higgins MS, Stewart R. 1962. Radiation stress and mass transport in gravitywaves with application to surf beats. J. Fluid Mech. 13:481504

52 Monismith


17/21

Lowe RJ, Falter JL, Bandet MD, Pawlak G, Atkinson MJ, et al. 2005a. Spec-tral wave dissipation over a barrier reef. J. Geophys. Res. (Oceans) 110:

doi:10.1029./2004.JC002711Lowe RJ, Koseff JR, Monismith SG. 2005b. Oscillatory flow through sub-

merged canopies. Part 1. Velocity structure. J. Geophys. Res. (Oceans)110:doi:10.1029./2004.JC002788

Lowe RJ, Monismith SG, Koseff JR, Falter J. 2005c. Oscillatory flow through sub-merged canopies. Part 2. Canopy mass transfer.J. Geophys. Res. (Oceans)110:doi:

10.1029./2004.JC002789Lueck RG, Lu Y. 1998. The logarithmic layer in a tidal channel. Cont. Shelf Res.

17:17851801Lugo-Fernandez A, Roberts HH, Wiseman WJ. 1998a. Tide effects on wave attenu-

ation and wave set-up on a Caribbean coral reef.Est. Coast. Shelf. Sci.47:38593Lugo-Fernandez A, Roberts HH, Wiseman WJ, Carter BL. 1998b. Water level and

currents of tidal and infragravity periods at Tague Reef, St. Croix (USVI). Coral

Reefs17:34349Massel SR, Gourlay MR. 2000. On the modeling of wave breaking and set-up on

coral reefs.J. Coastal Eng. 39:127

McDonald CB, Koseff JR, Monismith SG. 2006. Effects of the depth to coral heightratio on drag coefficients for unidirectional flow over coral: a reconciliation of

current data.Limnol. Oceanog.In pressMei CC. 1989.Applied Dynamics of Ocean Surface Waves. Singapore: World Sci. Press.

740 pp.Monismith SG, Genin A, Reidenbach MA, Yahel G, Koseff JR. 2006. Thermally

driven exchanges between a coral reef and the adjoining ocean. J. Phys. Ocean.

36:133247Munk WM, Ewing GC, Revelle RR. 1949. Diffusion in Bikini Lagoon.Trans. Am.

Geophys. Union30:15966Munk WH, Sargent MC. 1948. Adjustment of Bikini atoll to ocean waves. Trans. Am.

Geophys. Union29:85560Nakamura T, van Woesik R. 2001. Water-flow rates and passive diffusion partially

explain differential survival of corals during the 1998 bleaching event.Mar. Ecol.

Prog. Ser. 212:30104Neiman H, Richter C, Jonkers H, Badran MI. 2004. Red Sea gravity currents cascade

near-reef phytoplankton to the twilight zone.Mar. Ecol. Prog. Ser. 269:9199Niiler PP. 1968. On the internal tidal motions in the Florida Straits. Deep Sea Res.

15:11323Odum HT, Odum EP. 1955. Trophic structure and productivity of a windward coral

reef community on Eniwetok Atoll.Ecol. Monog. 25:291320Prager EJ. 1991. Numerical simulation of circulation in a Caribbean-type backreef

lagoon.Coral Reefs10:17782Raupach MR, Thom AS. 1981. Turbulence in and above plant canopies. Annu. Rev.

Fluid Mech. 13:97121Reidenbach MA, Koseff JR, Monismith SG, Steinbuck JV, Genin A. 2006a. Effects

of waves, unidirectional currents, and morphology on mass transfer in branchedreef corals.Limnol. Oceanogr.51: 113441



18/21

Reidenbach MA, Monismith SG, Koseff JR, Yahel G, Genin A. 2006b. Boundarylayer turbulence and flow structure over a fringing coral reef. Limnol. Oceanogr.51:195668

Riegl B, Piller WE. 2003. Possible refugia for reefs in times of environmental stress.

Int. J. Earth Sci. (Geol. Rundsch). 92:52031

Rinkevich B, Loya Y. 1983. Oriented translocation of energy in grafted reef corals.

Coral Reefs1:24347

Roberts HH, Murray SP, Suhayda JH. 1975. Physical process in a fringing reef

system.J. Mar. Res. 33:23360

Roberts HH, Suhayda JN. 1983. Wave-current interactions on a shallow reef(Nicaragua, Central America).Coral Reefs1:20914

Roberts HH, Wilson PA, Lugo-Fernandez A. 1992. Biologic and geologic responses

to physical proceses: examples from modern reef systems of the Caibbean-

Atlantic region. Cont. Shelf Res. 12:80934

Sebens KP, Vandersall KS, Savina LA, Graham KR. 1996. Zooplankton capture by

two scleractinian corals, Madracis mirabilisand Montastrea cavernosa, in a field

enclosure.Marine Biol. 127:30317

Skotner C, Apelt CJ. 1999. Application of a Boussinesq model for the computation ofbreaking waves. Part 2: wave-induced setdown and setup on a submerged coral

reef.Ocean. Eng. 26:92747

Sturman JJ, Oldham CE, Ivey GN. 1999. Steady convective exchange down slopes.

Aquatic Sci. 61:26078

Symonds G, Gardiner-Garden R. 1994. Coastal density currents forced by cooling

events.Cont. Shelf Res. 14:14357

Symonds G, Black KP, Young IR. 1995. Wave-driven flow over shallow reefs.

J. Geophys. Res. 100:263948

Symonds G, Black KP. 2001. Predicting wave-driven currents on surfing reefs.

J. Coastal Res. 29:10214

Tait RJ. 1972. Wave set-up on coral reefs. J. Geophys. Res. 77:220711

Tartinville B, Deleersnijder E, Rancher J. 1997. The water residence time in the

Mururoa atoll lagoon: sensitivity analysis of a three-dimensional model. Coral

Reefs16:19303

Tartinville B, Rancher J. 2000. Wave-induced flow over Mururoa Atoll Reef.J. Coast.

Res. 16:77681

Thomas FIM, Atkinson MJ. 1997. Ammonium uptake by coral reefs: effects of water

velocity and surface roughness on mass transfer.Limnol. Oceanogr. 42:8188Veron JEN, Pichon V. 1976. Scleractinian of eastern Australia, Vol. I: Families Tham-

nasteriidae, Astrocoeniidae, Pocilloporidae. InAustralian Institute of Marine Sci-ence Monograph 1. Townsville, Australia

Vogel JC, Eaton JK. 1985. Combined heat transfer and fluid dynamic measurements

downstream of a backward-facing step. J. Heat Trans. 107:92229

Von Arx WS. 1954. Circulation systems of Bikini and Rongelap lagoons, Bikini and

nearby atolls, Marshall Islands.US Geol. Surv. Prof. Pap.260-B: 26573

Wolanski E. 1983. Tides on the northern Great Barrier Reef continental shelf.

J. Geophys. Res. 88:595359

54 Monismith


19/21

Wolanski E. 1987. Some evidence for boundary mixing near coral reefs. Limnol.

Oceanogr. 32:73539

Wolanski E, Aseada T, Tanaka A, Deleersnijder E. 1994. Three-dimensional island

wakes in the field, laboratory and numerical models. Cont. Shelf Res. 156:143752

Wolanski E, Bennett AF. 1983. Continental shelf waves and their influence on the

circulation around the great barrier reef. Aust J. Mar. Freshwater Res. 34:2347

Wolanski E, Imberger J, Heron M. 1984. Island wakes in coastal waters.J. Geophys.

Res. 89:1053369

Yahel G, Post AF, Fabricius K, Marie D, Vaulot D, Genin A. 1998. Phytoplankton

distribution and grazing near coral reefs. Limnol. Oceanogr. 4:55163Young IR. 1989. Wave transformations on coral reefs. J. Geophys Res. 94:997989



20/21

Annual Review

Fluid Mechani

Volume 39, 200

Contents

H. Julian Allen: An Appreciation

Walter G. Vincenti, John W. Boyd, and Glenn E. Bugos 1

Osborne Reynolds and the Publication of His Papers

on Turbulent Flow

Derek Jackson and Brian Launder 18

Hydrodynamics of Coral Reefs

Stephen G. Monismith 37

Internal Tide Generation in the Deep Ocean

Chris Garrett and Eric Kunze 57

Micro- and Nanoparticles via Capillary Flows

Antonio Barrero and Ignacio G. Loscertales 89

Transition Beneath Vortical Disturbances

Paul Durbin and Xiaohua Wu 107

Nonmodal Stability Theory

Peter J. Schmid 129

Intrinsic Flame Instabilities in Premixed and Nonpremixed

Combustion

Moshe Matalon 163

Thermofluid Modeling of Fuel Cells

John B. Young 193

The Fluid Dynamics of Taylor Cones

Juan Fernndez de la Mora 217

Gravity Current Interaction with Interfaces

J. J. Monaghan

245

The Dynamics of Detonation in Explosive Systems

John B. Bdzil and D. Scott Stewart 263

The Biomechanics of Arterial Aneurysms

Juan C. Lasheras 293

vii


21/21

The Fluid Mechanics Inside a Volcano

Helge M. Gonnermann and Michael Manga

Stented Artery Flow Patterns and Their Effects on the Artery Wall

Nandini Duraiswamy, Richard T. Schoephoerster, Michael R. Moreno,

and James E. Moore, Jr.

A Linear Systems Approach to Flow Control

John Kim and Thomas R. Bewley

Fragmentation

E. Villermaux

Turbulence Transition in Pipe Flow

Bruno Eckhardt, Tobias M. Schneider, Bjorn Hof, and Jerry Westerweel

Waterbells and Liquid Sheets

Christophe Clanet

Indexes

Subject Index

Cumulative Index of Contributing Authors, Volumes 139

Cumulative Index of Chapter Titles, Volumes 139

Errata

An online log of corrections toAnnual Review of Fluid Mechanicschapters

(1997 to the present) may be found at http://fluid.annualreviews.org/errata.shtml

Hydrodynamic of reefs

Documents

Transcript of Hydrodynamic of reefs